Abstract

We demonstrate quantitative phase delay measurements with a spiral phase contrast microscope working in confocal mode. Such a confocal configuration is sensitive to weak phase objects due to background rejection but does not give direct access to the phase delay introduced by the sample. We develop a theory showing that shifting the illumination spot relative to the detector gives access to the local phase gradient in the first-order approximation. Subsequently, we present an iterative integration algorithm for phase delay measurements. This approach is validated on simulated and calibrated experimental images. Finally, the algorithm is applied to measure the phase profile of a cell, in which phase delays of 10 mrad are observed.

© 2014 Optical Society of America

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  1. R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
    [CrossRef]
  2. P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
    [CrossRef]
  3. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
    [CrossRef]
  4. G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
    [CrossRef]
  5. M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
    [CrossRef]
  6. C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
    [CrossRef]
  7. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
    [CrossRef]
  8. S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
    [CrossRef]
  9. D. K. Hamilton and T. Wilson, “Two-dimensional phase imaging in the scanning optical microscope,” Appl. Opt. 23, 348–352 (1984).
    [CrossRef]
  10. M. R. Atkinson and A. E. Dixon, “Single-pinhole confocal differential phase contrast microscopy,” Appl. Opt. 33, 641–653 (1994).
    [CrossRef]
  11. W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
    [CrossRef]
  12. G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
    [CrossRef]
  13. C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
    [CrossRef]
  14. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express 16, 16240 (2008).
    [CrossRef]
  15. N. Lue, W. Choi, K. Badizadegan, R. R. Dasari, M. S. Feld, and G. Popescu, “Confocal diffraction phase microscopy of live cells,” Opt. Lett. 33, 2074–2076 (2008).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
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    [CrossRef]
  18. C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
    [CrossRef]
  19. R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
    [CrossRef]
  20. I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
    [CrossRef]
  21. S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.
  22. P. Török and P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
    [CrossRef]
  23. E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  24. A. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
    [CrossRef]
  25. J. Pawley, ed., Confocal Microscopy with Transmitted Light (Plenum, 1995).

2013 (1)

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

2011 (2)

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

2010 (1)

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

2009 (1)

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

2008 (2)

2006 (2)

2005 (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

2004 (1)

2003 (1)

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

1999 (1)

1995 (1)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

1994 (1)

1993 (1)

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

1992 (1)

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

1984 (1)

1981 (1)

1979 (1)

G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1952 (1)

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef]

Amos, W. B.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Atkinson, M. R.

Badizadegan, K.

Barer, R.

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Bevilacqua, F.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Boss, D.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Brakenhoff, G.

G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
[CrossRef]

Cattermole, D. M.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Choi, W.

Cogswell, C.

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

Cuche, E.

Dasari, R. R.

Depeursinge, C.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

Devaney, A.

Dixon, A. E.

Elias, L.

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Emiliani, V.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Feld, M. S.

Fürhapter, S.

Gase, R.

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

Guillon, M.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Hamilton, D. K.

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Jourdain, P.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Kimel, I.

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Laufer, J.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Lauterbach, M. A.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Liu, C.-L.

S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.

Lue, N.

Magistretti, P. J.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Marquet, P.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

Moratal, C.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Munro, P. R. T.

Osten, W.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Pavillon, N.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Pedrini, G.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Pei, S.-C.

S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.

Popescu, G.

Rappaz, B.

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

Reichelt, S.

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23, 1400–1409 (2006).
[CrossRef]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Sheppard, C.

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

Situ, G.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Soltani, A.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Török, P.

Warber, M.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Wilson, T.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

I. Kimel and L. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

J. Microsc. (3)

W. B. Amos, S. Reichelt, D. M. Cattermole, and J. Laufer, “Re-evaluation of differential phase contrast (DPC) in a scanning laser microscope using a split detector as an alternative to differential interference contrast (DIC) optics,” J. Microsc. 210, 166–175 (2003).
[CrossRef]

G. Brakenhoff, “Imaging modes in confocal scanning light-microscopy (CSLM),” J. Microsc. 117, 233–242 (1979).
[CrossRef]

C. Cogswell and C. Sheppard, “Confocal differential interference contrast (DIC) microscopy—including a theoretical-analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

J. Neurosci. (1)

P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31, 11846–11854 (2011).
[CrossRef]

J. Opt. A (1)

C. Maurer, S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral phase Fourier filter,” J. Opt. A 11, 094023 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser Photon. Rev. (1)

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[CrossRef]

Nature (1)

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef]

Opt. Commun. (2)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Opt. Commun. 283, 1273–1277 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef]

Sci. Rep. (1)

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013).
[CrossRef]

Other (3)

J. Pawley, ed., Confocal Microscopy with Transmitted Light (Plenum, 1995).

S.-C. Pei and C.-L. Liu, “A general form of 2D Fourier transform eigenfunctions,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 3701–3704.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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